On the Frobenius fields of abelian varieties over number fields

Let \(A\) be a non-CM simple abelian variety over a number field \(K\). For a place \(v\) of \(K\) such that \(A\) has good reduction at \(v\), let \(F(A,v)\) denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming \(A\) has connected monodromy groups, we show that the set of places \(v\) such that \(F(A,v)\) is isomorphic to a fixed number field has upper Dirichlet density zero. Assuming the GRH, we give a power saving upper bound for the number of such places.